# Poincaré recurrence theorem

Let $(X,\mathcal{S},\mu )$ be a probability space^{} and let $f:X\to X$
be a measure preserving transformation.

###### Theorem 1.

For any $E\mathrm{\in}\mathrm{S}$, the set of those points $x$ of $E$ such that ${f}^{n}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{\notin}E$ for all $n\mathrm{>}\mathrm{0}$ has zero measure. That is, almost every point of $E$ returns to $E$. In fact, almost every point returns infinitely often; i.e.

$$\mu \left(\{x\in E:\text{there exists}N\text{such that}{f}^{n}(x)\notin E\text{for all}nN\}\right)=0.$$ |

The following is a topological version of this theorem:

###### Theorem 2.

If $X$ is a second countable Hausdorff space and $\mathrm{S}$ contains the Borel sigma-algebra, then the set of recurrent points of $f$ has full measure. That is, almost every point is recurrent.

Title | Poincaré recurrence theorem |
---|---|

Canonical name | PoincareRecurrenceTheorem |

Date of creation | 2013-03-22 14:29:50 |

Last modified on | 2013-03-22 14:29:50 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 37B20 |

Classification | msc 37A05 |